For future reference, we state [1]}. For \(I\)  a set of weights, we denote by \(\sigma _I\)  the sum of the weights in \(I\) . Let \(\chi \)  be a weight in \(M\) . Let \(\chi ^+\)  be the dominant Weyl-shifted conjugate of \(\chi \)  if it exists, and zero otherwise. Let
\(w\)  be the element of the Weyl group such that \(w*(\chi -\sigma _I)\)  is dominant or zero. It has length \(\ell (w)=:\ell (I)\) .
